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Special Topics

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Abstract

This chapter addresses some special topics which are particularly relevant for current research involving cold atoms and molecules. Section 4.1 on deep potentials falling off faster than \(1/r^{ 2}\) at large distances contains a general theory for the description of near-threshold bound and continuum states, which is well suited for application to realistic binary systems such as diatomic molecules or molecular ions. Section 4.2 connects the well established theory of Feshbach resonances with the empirical description which has become widely used in the cold-atoms community, and it formulates a threshold-insensitive parametrization of the Feshbach resonances which is relevant for the analysis of current experiments. The last section contains a short treatise on two-dimensional scattering, which reveals significant differences to the 3D case, in particular in the low-energy, near-threshold regime.

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Notes

  1. 1.

    Noninteger values of \(l\) are not merely of academic interest. They can describe the effects of inverse-square potentials of other origin than the centrifugal term. In two-dimensional scattering described in Sect. 4.3, the radial Schrödinger equation with integer angular momentum quantum number \(m\) resembles that of the 3D case when \(l=\vert m\vert-{1\over 2}\).

  2. 2.

    Due to the \(m\)-independent term \({\pi\over 4}\) appearing in the arguments both of \(u_{m}^{(\mathrm{s})}(kr)\) and of \(u_{m}^{(\mathrm{c})}(kr)\) in (4.217), there is no a priori preference for the assignment of an asymptotic “sine-” or “cosine-like” behaviour. The present nomenclature is chosen to make the connection to the 3D case as transparent as possible.

  3. 3.

    Since \(m=0\) corresponds to \(l=-{1\over 2}\), the phase shift \(\tilde{\delta}\) in Sect. 2.7.2.3 is actually the scattering phase shift \(\delta_{m=0}\) in the present case, see Eq. (2.341).

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    Harald Friedrich

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Friedrich, H. (2016). Special Topics. In: Scattering Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48526-2_4

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